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of the five string theories. There are

known to be at least 10-100,000 families

of these manifolds. Each family can have

a dimension of order 100 or more, i.e. comes

with hundreds of parameters which in some

sense determine the size and shape of the

Calabi-Yau.

It is quite possible the number of Calabi

Yau families is infinite. This remains an

open problem in algebraic geometry and you

can find some algebraic geometers who

believe the number is finite, some who believe it is infinite.

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I was thinking also if could be some way of decompactify the dimensions trapped in a CY manifold--maybe manipulating it somehow?

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the universe starts off with 10 flat dimensions

then 6 get compactified. As far as anyone

knows, if string theory makes any sense, strings in 10 flat dimensions makes sense,

and there is no mechanism (other than

wishful thinking) for six dimensions to spontaneously compactify.

As far as anyone knows, one Calabi-Yau

is as good as any other. In 1984, people

hoped there was a small number of Calabi-Yaus and maybe some way to rule out

all but a unique one. Nobody believes this

particular piece of wishful thinking anymore.

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I think that you are not understanding what's the gist of what I'm asking (my blame too, cause I'm not very comfortable writting in english). The question is: Did the dimensions compactified in one (a single, a unique) CY manifold, so they are compactified in THIS manifold that is located somewhere, or did they compactified in more than one? Is like asking: If we have the volume of the universe,say 10^{100} km^{3}, and then if there's only ONE CY manifold in this volume then the density of CY manifolds is 1/(10^{100} km^{3}), or perhaps, there are 100 CY manifolds then the density is 100/(10^{100} km^{3}). This is the type of information that I want to know: the number (nor the different types) of CY manifolds that are predicted to exist

I've read that 6 dimensions were compactified starting from an initial state of 10 dimensions:

http://www.fortunecity.com/emachines/e11/86/dimens.html [Broken]"

"But, of course, all this takes place in 10 dimensions. Physicists retrieve our more familiar 4-dimensional Universe by assuming that, during the big bang, 6 of the 10 dimensions curled up (or "compactified")"

I've read that 6 dimensions were compactified starting from an initial state of 10 dimensions:

http://www.fortunecity.com/emachines/e11/86/dimens.html [Broken]"

"But, of course, all this takes place in 10 dimensions. Physicists retrieve our more familiar 4-dimensional Universe by assuming that, during the big bang, 6 of the 10 dimensions curled up (or "compactified")"

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all to say about your question. It has nothing

to say about how compactification comes

about, and thus nothing to say about which

CY manifolds or how many you end up with.

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Here is a great paper I just found on the mathematics of C-Y manifolds.:

http://www-thphys.physics.ox.ac.uk/users/PeterAusting/Conference/Talks/candelas.pdf [Broken]

The best book is Tristan Hubsch's Calabi-Yau Manifolds: A Bestiary for Physicists.

http://www-thphys.physics.ox.ac.uk/users/PeterAusting/Conference/Talks/candelas.pdf [Broken]

The best book is Tristan Hubsch's Calabi-Yau Manifolds: A Bestiary for Physicists.

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"Whatever definition is used, Calabi-Yau manifolds, as well as their moduli spaces, have interesting properties. One is the symmetries in the numbers forming the Hodge diamond of a compact Calabi-Yau manifold. It is surprising that these symmetries, called mirror symmetry, can be realized by another Calabi-Yau manifold, the so-called mirror of the original Calabi-Yau manifold. The two manifolds together form a mirror pair. Some of the symmetries of the geometry of mirror pairs have been the object of recent research."

http://mathworld.wolfram.com/Calabi-YauSpace.html

BTW, the concept of mirror symmetry was introduced by Brian Greene

So, I will try to know what's that Hodge diamond. So, if two Cy manifolds form a mirror pair, I guess that must be two ADJACENT CY manifolds, or perhaps two CY manifolds that are separated certain distance can also form a mirror pair?

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"the hodge diamond displays the Hodge numbers of a manifold"

from here

http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_diamond/hodge_diamond.html [Broken]

Hodge number: "The Hodge number is the analog of the Betti number on real manifolds for complex manifolds"

This is the definition in this page

http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_number/hodge_number.html#hodge [Broken] number

from here

http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_diamond/hodge_diamond.html [Broken]

Hodge number: "The Hodge number is the analog of the Betti number on real manifolds for complex manifolds"

This is the definition in this page

http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_number/hodge_number.html#hodge [Broken] number

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